There are two basic directions on the lattice: multiplication and division.

If I start with a note, and then multiply it by 3, or 5, or 7, I will get a harmony note with overtonal energy. Such a note is in the natural overtone series of the original note.

Overtonal energy is stable, restful, it belongs where it is and wouldn’t mind staying there.

If I divide by 3, 5 or 7, I get a completely different kind of note. I call this division energy “reciprocal,” after W.A. Mathieu’s suggestion in his amazing book Harmonic Experience.

Reciprocal energy is restless, unstable. The note wants to move, or for the music to come to it, until it is overtonal.

On the lattice of fifths and thirds, there are two axes, fifths and thirds, and two directions, overtonal and reciprocal.

This makes four total directions one can move on this lattice. Each direction has own characteristic flavor, or energy. I use the following names for these energies, mostly after Mathieu.

Dominant = East = Overtonal fifths

Subdominant = West = Reciprocal fifths

Major = North = Overtonal thirds

Minor = South = Reciprocal thirds

Every interval has its own unique recipe of moves in these four directions. The perfect fifth has pure dominant energy, the major third pure major. The minor third, b3 on the lattice, is a compound note — dominant and minor.

It’s interesting to look at the minor third (b3) from the viewpoint of tonal gravity. On the horizontal axis, dominant/subdominant, the b3 is overtonal, stable, restful. On the vertical axis, major/minor, the note is reciprocal, unstable, restless.

Tonal gravity is stronger the closer you are to the center. To make a minor third, you multiply by 3 (an overtonal jump of a fifth), and divide by 5 (a reciprocal jump of a third). I know, 3 generates fifths and 5 generates thirds, a confusing coincidence.

Fifths are closer to the center, harmonically, than thirds are, so the overtonal energy is stronger than the reciprocal.

This makes the minor third a stable note, although less stable than the major third. Songs can end on a tonic minor chord and they will still sound finished.

The photo to the right was taken by the Cassini spacecraft when the sun was behind the planet, and backlighting the rings and the edge of the atmosphere. A type of solar eclipse never before seen by humans!

The rings are composed of millions of small particles, mostly ice, orbiting Saturn. They are neither arranged in a uniform disc, nor are they evenly spaced like the grooves on a record. Instead, they have an exquisite natural pattern, not quite like anything I know of on Earth. Click the image below for a full size, zoomable version. The bright bands are higher density, dark ones lower density.

As with so many patterns in nature, this one is generated by simple rules. The main generator of the ring patterns is orbital resonance. The chunks that form the rings come in all sizes, from dust grains to small moons. When the orbital periods of two bodies are related to each other by a ratio of small whole numbers (sound familiar?), they will have a lot more gravitational influence on each other, just like the playground swing example from the last post. They give each other a little kick every time they come around, and either the relationship is unstable (one or both get booted out of their orbit) or stable (they settle in to a pattern and their resonance locks them into, um, harmony).

There are other examples in the Solar System. Pluto and Neptune are in a 3:2 resonance. Pluto orbits the sun twice for every three times Neptune goes around, and the relationship has persisted for a long time. They are playing a very slow perfect fifth. Orbital resonance draws them into this pattern. The legs are kicking at just the right time.

I think there is a very real connection between the beauty of the rings and the beauty of harmony. Stand close to someone, and sing a note while the other person sings a perfect fifth above. I think you will feel the resonance in your vocal cords, as it draws you into entrainment. Resonance influences and creates physical structures on every scale from subatomic particles to spiral galaxies.

Once again, I propose that when we experience the joy of musical harmony, we are seeing (and hearing and feeling) a little more deeply into the nature of the universe. The window is resonance. Here’s an interesting site with lots more about the connections between physics, sound and resonance.

And, to ride my hobby horse for just a second, I believe the dominance of equal temperament has obscured this deep insight and feeling. For many notes, the legs just don’t kick at quite the right time. No worries, I do think equal temperament is extremely useful, and it’s been used to make a whole lot of gorgeous music. I use it myself. But it has distanced us somewhat from the shot of pure joy that the resonances of music, in tune, can deliver. I’m hooked on the straight stuff, and the reason I’m writing this blog is the desire to share that joy.

Actually, I do know of an Earthly structure that resembles Saturn’s rings. It’s the scale, in just intonation.

Resonance, in general, is the tendency of a system to vibrate more strongly at some frequencies than at others.

A great example of this is a playground swing. Like any pendulum, the swing “wants” to swing at a particular frequency, its resonant frequency. You can make yourself go higher and higher without a lot of effort — if you swing your legs at just the right time. Move your legs a little too fast, or two slow, and you won’t go any higher. You can stop yourself cold, just by swinging your legs at a “wrong” frequency.

A similar thing happens when you sing in the shower. If you run up and down the scale, you will find that some frequencies are reinforced, and some canceled. It is much easier to make a louder sound if you are singing at the shower’s resonant frequency. Try it next time — pick your favorite shower song (One of mine is Englishman in New York — be yourself, no matter what they say) and sing up and down until you find a note that easily gets really loud. Then sing the song in that key.

I recently read Mickey Hart’s beautiful book, Drumming at the Edge of Magic. Hart is best known as one of the two drummers of the Grateful Dead. His book tells the story of his lifelong fascination with percussion, and of his investigations into the ancient connection between rhythm and the human spirit.

Toward the end of the book, Hart introduced a concept that was new to me — entrainment.

It seems that if two vibrating systems are allowed to interact, and if their frequencies are already somewhat close to each other, they will become synchronized. The faster one slows down, and the slow one speeds up. Here’s a demonstration:

One of the greatest joys in my life is making music with other people. It’s great to play solo, but something different happens when two or more people make music together. I think a part of the reason is that the musicians, and often the audience, entrain with each other. They each keep their own time, and it won’t be identical at first. But as they listen to each other, and feel the common rhythms, their grooves start to adjust until they are literally on the same wavelength, and something happens inside them, one of the flavors of ecstasy. It is a bonding experience, and I feel a different connection with anyone I’ve entrained with in this way.

The beauty of this is that it’s a physical principle, which means it’s not something you have to “try” to do. Just relax and pay attention to what the other band members are doing, and if you are in the same room and have a chance to influence each other, the laws of nature will pull you toward entrainment. Do it long enough and everyone involved, including the listeners, is likely to experience a trance state. Hart should know, the Dead accomplished this thousands of times over their long career.

In his book, Mickey Hart talked about entrainment in the context of rhythm. But I immediately connected it with harmony. I sang Christmas carols professionally for years, with the Dickens Carolers quartets in Seattle. When all four voices blend perfectly, there is a delicious sensation of being a single voice. Once, four of us were walking back to the car from a show, and we ran across the Kobe Bell, a landmark feature of Seattle Center. We all stood with our heads up inside the bell, found a key that resonated with it, and sang O Holy Night. It was one of the peak experiences of my life. The bonding that occurred was almost frightening.

I think that singing harmony is like dancing together, only very fast.